Defining an appropriate range of scales for application of the gradient Richardson number, with implications for observations of stratified shear turbulence at laboratory and ocean scales

The Richardson number ( Ri ) represents the square of the ratio of buoyancy frequency to vertical shear, and a value less than ¼ has long been recognized as a necessary condition for the generation of turbulence, particularly at small scales. At larger scales, it is common to evaluate a bulk Richardson number, ( Ri B ), with arbitrary values of criticality, generally greater than ¼. Despite the ubiquity of this concept in modern oceanography, the range of scales over which the critical value of ¼ is valid has not been well documented. Here, spectral and energetics arguments are used to identify the primitive shear length scale, l S = ( ν S ¯ ) 1 2 , where ν is kinematic viscosity and S ¯ is velocity shear, as the fundamental scale for appropriate scales for a critical value of Ri = ¼. These findings are evaluated against a variety of data, suggesting that the range 10 ls – 100 ls is an approximate range of validity for the critical value of ¼. This range is equivalent to 100 Re 10,000, where Re is a Reynolds number based on thickness and velocity across the layer. Further data analysis suggests that turbulence persists at values of Ri B far above ¼ in very thick layers ( Re 10 5 ), and that turbulence intensity is enhanced for thin layers ( Re 10 5 ). We hypothesize the former is due to the existence of smaller scale regions within the layer where Ri locally falls below ¼, while the latter is the result of stratification increasing the ratio of turbulent kinetic energy to fluid mass. Historically, the value of Re has been considered of minimal relevance to stratified turbulence in the ocean. However, here we demonstrate the significance of Re , and suggest a broader view of turbulence within Re - Ri parameter space. The proposed parameter space may potentially yield new insights into turbulence closures, particularly at lower Re values, provide a more rigorous approach to defining critical values of Ri B , greatly facilitating interpretation of observational data, and provide a more rigorous framework for the use of direct numerical simulation (DNS) results and laboratory experiments of turbulence to inform geophysical scale dynamics.